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Theorem r0cld 22321
Description: The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
r0cld ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 22308 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1130 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 Fn 𝑋)
4 fncnvima2 6804 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
53, 4syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
6 fvex 6656 . . . . . 6 (𝐹𝑧) ∈ V
76elsn 4555 . . . . 5 ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ (𝐹𝑧) = (𝐹𝐴))
8 simpl1 1188 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 simpr 488 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝑧𝑋)
10 simpl3 1190 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐴𝑋)
111kqfeq 22307 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
12 eleq2w 2895 . . . . . . . . 9 (𝑦 = 𝑜 → (𝑧𝑦𝑧𝑜))
13 eleq2w 2895 . . . . . . . . 9 (𝑦 = 𝑜 → (𝐴𝑦𝐴𝑜))
1412, 13bibi12d 349 . . . . . . . 8 (𝑦 = 𝑜 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑜𝐴𝑜)))
1514cbvralvw 3426 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜))
1611, 15syl6bb 290 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
178, 9, 10, 16syl3anc 1368 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
187, 17syl5bb 286 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
1918rabbidva 3455 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}} = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
205, 19eqtrd 2856 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
211kqid 22311 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22213ad2ant1 1130 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23 simp2 1134 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ Fre)
24 simp3 1135 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐴𝑋)
25 fnfvelrn 6821 . . . . . 6 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
263, 24, 25syl2anc 587 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
271kqtopon 22310 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28273ad2ant1 1130 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29 toponuni 21497 . . . . . 6 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3028, 29syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → ran 𝐹 = (KQ‘𝐽))
3126, 30eleqtrd 2914 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ (KQ‘𝐽))
32 eqid 2821 . . . . 5 (KQ‘𝐽) = (KQ‘𝐽)
3332t1sncld 21909 . . . 4 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝐴) ∈ (KQ‘𝐽)) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
3423, 31, 33syl2anc 587 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35 cnclima 21851 . . 3 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3622, 34, 35syl2anc 587 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3720, 36eqeltrrd 2913 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  {crab 3130  {csn 4540   cuni 4811  cmpt 5119  ccnv 5527  ran crn 5529  cima 5531   Fn wfn 6323  cfv 6328  (class class class)co 7130  TopOnctopon 21493  Clsdccld 21599   Cn ccn 21807  Frect1 21890  KQckq 22276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-qtop 16758  df-top 21477  df-topon 21494  df-cld 21602  df-cn 21810  df-t1 21897  df-kq 22277
This theorem is referenced by:  nrmr0reg  22332
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